What is the fastest GPU possible without bottlenecking this CPU?

What is the fastest GPU setup you can have without encountering any major bottlenecks with an AMD Phenom II X4 965BE? Is it absolutely imperative to have a Core i7 clocked @ 4.0 GHz to properly run the latest graphics cards, especially in SLI/CrossfireX?

My monitor only goes up to a 720p resolution. I heard that a bottleneck only appears at a certain resolution.

Even if I have a bottleneck with my new PC, should I still see a huge performance increase across every graphical setting, compared to my old PC and HD 4850?

The HD 5870 is almost Three times faster than my HD 4850. Most games are GPU bound, rather than CPU bound. A bottleneck with a Phenom II X4 965 @ stock speeds won't pose a crippling hit to performance with an HD 5870?

The GPU bottlenecking the CPU or vice-versa phenomenon is a myth. Although it's possible in practice, it is an atypical situation regardless of virtually any reasonable system specs.

A sufficiently slow CPU will limit the framerates you can obtain in a game, but this is not a GPU bottleneck, it is merely a bottleneck in the CPU-part of code execution. It's the CPU bottlenecking itself, the GPU can still run as fast as it pleases regardless of CPU insufficiency.

A slow CPU is evident at lower resolutions rather than higher. At higher resolutions you increase the GPU part of the work load. Only when the GPU is relieved of work does the performance difference between CPUs become more evident.

There is a very real limit on the bandwidth of the PCIe bus, and there is a limit to how much data a CPU can pipe through a bus. You can run into circumstances where bandwidth across the bus and/or the processor's ability to pipe information is the limiting factor in an application, rather than arithmetic performance or local latency/bandwidth on the graphics card, however this is a very rare issue in a well developed game. A properly built game should exhaust the arithmetic performance of the GPU regardless of bandwidth constraints, so even on an extremely slow CPU and bus, the GPU does not need to be constrained. Your limiting factor at high resolutions will be video memory. If your video memory is insufficient, the CPU will need to do more work to perform constant swapping, however in this case a faster CPU will merely bandage the source of the problem. Latency to data will be dramatically increased and no CPU of any magnitude will help. The golden rule is simply to have enough video memory. If you have that, your CPU can go on vacation as far as the GPU is concerned.

This may all seem theoretical but it is generally true in practice in modern games. In fact in the vast majority of cases you can go from a middle of the road Core 2 Duo to a top of the line Core i7 Extreme and you won't witness any meaningful performance improvement, if any improvement at all, regardless of your graphics devices. Core i7s have a habit of occasionally performing worse than their predecessors at high gaming resolutions.

There is no realistic limit to how fast a GPU/GPUs you can put in there. Two 5970s would work without concerns. I would also strongly recommend against the Phenom. The only justifiable upgrade to your system as it is right now is to give it as many GPUs as you can afford. You would without question see far bigger gains in performance pushing the CPU budget into the GPU to almost no end.

I intend to keep my Core 2 Duo on 4GB RAM and PCIe 1.0 until at least the 6800/GT500 generation of GPUs, and probably some years after. CPU upgrades have shown to make remarkably little sense in recent years.

So a Phenom II X4 965 shouldn't have any issues running a single Radeon HD 5870, and I should still notice a HUGE performance increase in all of my games compared to my HD 4850? With our without a bottleneck.

Your limiting factor at high resolutions will be video memory. If your video memory is insufficient, the CPU will need to do more work to perform constant swapping, however in this case a faster CPU will merely bandage the source of the problem.

A properly built game should exhaust the arithmetic performance of the GPU regardless of bandwidth constraints, so even on an extremely slow CPU and bus, the GPU does not need to be constrained.

That's debatable. Depends on resolution and settings. Every modern video game you can think of will be constrained by GPU if you run it in 2560x1600 with full antialiasing. The lower you go, the more the CPU and the bus matter. Older video games are more easily constrained by the CPU than newer video games. Something like Doom 3 or Call of Duty 2 in 1280x1024 resolution will reach the same 200 fps on the 4850 and the 5870.

That should be somewhat playable on the 4850 already. On the 5870 you should be getting 60-70 fps.

I can play STALKER CS on DX 10.1 on all "high" quality settings on 1024x768 and get >50fps without sun shadows activated. With sun shadows turned on, I get <15 fps.

It doesn't really matter, whatever game I play my PC powers itself off for no apparent reason after about 30-40 minutes. With the exception of a few, older, less demanding DX9 games like Battlefield 2, and Call of Duty 4.